The width tool will only change one line at a time as far as I know. There are, however, preset stroke forms that will change all selected paths at once.
With the select tool, select the paths you wish to change. Click the dropdown menu where it displays "Uniform" and select from there.
You can also create your own custom width "profile" using the width tool.
Select a path you wish to adjust. Use the width tool and form it the way you would like. Then, while the path is still selected, click the same dropdown I mentioned earlier (it won't say uniform anymore if you have manually adjusted the width). Then click the button labeled "Add to Profiles" and name your new width profile.
Once you set your new width profile, select all your paths you wish to change and select the new profile which will change all of them at once.